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## Lecture Topics

 Date What we discussed/How we spent our time Jan 14 Syllabus. Policies. Text. The main goals of the course are defined to be: (1) To learn what it means to say Mathematics is constructed to be well founded.'' To learn which concepts and assertions depend on which others. To learn what are the most primitive concepts ( = set, $\in$) and the most primitive assertions ( = axioms of set theory). (2) To learn how to unravel the definitions of function'', number'', and finite'', through layers of more and more primitive concepts, back to set'' and $\in$''. (3) To learn the meanings of truth'' and provability''. To learn proof strategies. (4) To learn formulas for counting. I will occasionally post notes for Math 2001 in the form of flash cards on Quizlet. To join our quizlet class, go https://quizlet.com/join/mExWGGZqj. (Test yourself on the Axioms of Set Theory with this Quizlet link: https://quizlet.com/_61ko6h.) Jan 16 We began discussing naive set theory alongside axiomatic set theory. We discussed the Axiom of Extensionality and the Axiom of the Empty Set. We introduced Venn diagrams, the directed graph representation of the universe of sets, and the concept of the successor of a set. (Some Venn diagrams from the web: 1, 2, 3, 4. The second one is not mathematically correct.) Jan 18 We discussed the Axioms of Infinity, Pairing, and Union. We defined the natural numbers to be the intersection of all inductive sets. Jan 23 First we defined subset and power set, and introduced the Axiom of Power Set. Then we turned to a discussion of abbreviations in mathematics. We defined the alphabet for set theory (variables, nonlogical symbols, logical symbols, punctuation), and how to write formal definitions for predicates. Worksheet 1 (+ solution sketches). Jan 25 We discussed the relationship between unrestricted comprehension and restricted comprehension. We showed, through Russell's Paradox, that the rule of unrestricted comprehension leads to a contradiction. We derived that there is no set of all sets. We also explained why there is no set containing all sets except one. Jan 28 We wrote the proof of the theorem $R=\{x\;|\;x\notin x\}$ is not a set'' in English. Then we discussed some of the history of the axioms of set theory, including the introduction of the Axiom of Replacement. (Because of the bad weather, and the fact that 30 percent of the class was absent, the Monday quiz was made into an ungraded practice worksheet.) Jan 30 Read pages 20-27. We discussed the Axiom of Choice and the Axiom of Regularity/Foundation. In the discussion of the Axiom of Foundation we defined the notion of an $\in$-minimal element of a set. We defined ZFC (all 10 axioms) and ZF (all 10 axioms minus the Axiom of Choice). We discussed classes (like the class of all sets), explaining what classes are and how they may differ from sets. We showed that the Axiom of Separation allows us to intersect any nonempty class of sets. Feb 1 Today we discussed De Morgan's Laws, $\overline{X\cap Y} = \overline{X} \cup \overline{Y}$ and $\overline{X\cup Y} = \overline{X} \cap \overline{Y}$. We also noted that $X\subseteq Y$ if and only if $\overline{X}\supseteq \overline{Y}$. Here $\overline{X}$ represents the complement of $X$ relative to some large set $A$. By referencing De Morgan's laws, we reasoned that $\cup$ and $\cap$ satisfy dual'' properties. As an example of this duality, we proved $X\subseteq X\cup Y$ directly, and then derived from this, by duality, that $X\supseteq X\cap Y$. (Quizlet allows me to write $X'$ but not $\overline{X}$, so on Quizlet I will write De Morgan's Laws as $(X\cap Y)' = X' \cup Y'$ and $(X\cup Y)' = X' \cap Y'$. Despite this duality, we noted that there are some asymmetries between union and intersection. The first asymmetry we noted was that $\bigcup \emptyset$ is a valid set (it is $\emptyset$), but $\bigcap \emptyset$ is not set. The second asymmetry we noted about union and intersection is that we can only form the union of a collection that is a set, but we can form the intersection of any collection that is a nonempty class. It is important that we can form these types of intersections, since the set of natural numbers, $\mathbb N$, is defined as the intersection of the class of inductive sets. (We explained today why the class of inductive sets is not a set.) Finally, we introduced the Kuratowski encoding of ordered pairs, namely $(a,b) := \{\{a\},\{a,b\}\}$. We stated that, with this definition the following theorem holds: Theorem. $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. (Not proved yet!) (Test yourself on set theory terminology with this Quizlet link: https://quizlet.com/_61ufo1. Some of these definitions are illustrated by examples here https://quizlet.com/_61vmmh.) Feb 4 We proved Theorem. $(a,b)=(c,d)$ if and only if $a=c$ and $b=d$. We defined ordered triple, ordered $n$-tuple, and Cartesian product $A\times B$. We explained why, if $A$ and $B$ are sets, then $A\times B$ is also a set. Quiz 1. Feb 6 Read pages 35-47. We defined relations and explained the connection between relations and predicates. Handout on relations. Feb 8 Read pages 28-35. We defined functions and went over some vocabulary for functions. Quiz yourself! Feb 11 Read pages 47-51. What type of mathematical object is a domain, codomain, image, or coimage? Any domain is a set and any set is a domain. Any codomain is a set and any set is a codomain. We defined the identity function on $S$, ${\rm id}_S:S\to S$, to show how to realize any set $S$ as a domain or a codomain. The image of a function is a subset of the codomain. Conversely, if $B$ is any set and $S\subseteq B$ is any subset, then there is a function, $\iota_S:S\to B$, the inclusion function for $S$ into $B$, for which $B$ is the codomain and $S$ is the image. The coimage of a function is a partition of the domain. Conversely, if $A$ is any set and $P$ is any partition of $A$, then there is a function, $N_P:A\to P: a\mapsto [a]$, the natural map for $P$, for which $A$ is the domain and $P$ is the coimage. Feb 13 Read pages 51-65. Directed graph representation of binary relations. Equivalence relations are the abstraction of kernels. Quiz yourself! Feb 15 What is a function? Recursion and induction, I. Exploiting the defining property of $\mathbb N$. Why is recursion a valid way to define a function? Why is induction a valid form of proof? Feb 18 Recursion and induction, II. Proving the laws of arithmetic. (Some hints.) Quiz 3. Feb 20 More induction proofs. Midterm Review Sheet. Feb 22 Cardinality, I. Finite versus infinite. Countable versus uncountable. Ordinal numbers. Cardinal numbers. Equipotence. Definitions of $|A|=|B|$, $|A|\leq|B|$, $|A|<|B|$. Cantor-Schroeder-Bernstein Theorem. Feb 25 Cardinality notes. Proof of Cantor-Schroeder-Bernstein Theorem. Proof of Cantor's Theorem. $|\mathbb N|<|{\mathcal P}(\mathbb N)|=|\mathbb R|=|\mathbb R^n|$. Quiz 4. Quiz yourself on ordinals and cardinals! Feb 27 Review for the midterm!. Mar 1 Midterm!. Mar 4 Read Subsections 4.1.1-4.1.2. First logic handout. Structures. Alphabet of symbols. Ingredients in a compound predicate. No quiz! Mar 6 Read Subsection 4.1.3. Second logic handout. We gave a recursive definition of the set of terms (or algebraic expressions) in some language. Then we discussed how to give meaning (= assign tables) to terms. Mar 8 Read Section 3.1. Third logic handout. Tables for the logical connectives. Tautology and contradiction. Logical equivalence of propositions. Quiz yourself on propositional logic! Mar 11 Fourth logic handout. Deciding the truth of a quantified statement. Quiz 5. Mar 13 Snow day! Campus closed! Mar 15 Fifth logic handout. We proved that every truth function equals a truth function written in disjunctive normal form. We defined complete set of connectives'', and listed some complete sets. Mar 18 Sixth logic handout. We defined atomic formulas and general formulas. We discussed how to standardize the variables apart'', and then how to put a formula in prenex form. Restricted quantifiers. Quiz 6. Mar 20 Practice with quantifiers! Mar 22 What is a proof? Axioms. Laws of Deduction, including Modus Ponens. Propositional tautologies versus logical tautologies versus logically valid sentences. Direct proof, proof of the contrapositive, and proof by contradiction. Proof by cases. Apr 1 Formal versus informal proof. We discussed the difference between semantic consequence ($\Sigma\models S$) and syntactic consequence ($\Sigma\vdash S$). Quiz 7. Apr 3 We discussed that the goal of any proof calculus is to be sound, complete, and decidable. We stated the Church-Turing Theorem about the undecidability of the set of logical validities, and Godel's Completeness Theorem about the the existence of a sound, complete, decidable proof calculus. We discussed the use of truth tables for designing proof strategies. Apr 5 Read Section 6.1. We proved the additive and multiplicative counting principles, using induction. Apr 8 We discussed counting independent events. We showed that the number of functions from $A$ to $B$ is $|B|^{|A|}$. We gave two proofs that $|{\mathcal P}(m)|=2^m$. We defined combinatorial proof. Quiz 8. Apr 10 We discussed this handout on distributions. We introduced the notations $P(n,k) = (n)_k = n^{\underline{k}}$ for $n!/(n-k)!$. Apr 12 Read Section 6.2. We defined the numbers $C(n,k) = {n \choose k}$ combinatorially. We showed that these numbers may be defined recursively. We showed that these are the numbers that arise in Pascal's Triangle. We showed that (i) the sum of the numbers in the $n$-th row of Pascal's triangle is $2^n$, (ii) the alternating sum of the numbers in the $n$-th row of Pascal's triangle is $0$ if $n>0$, (iii) Pascal's Triangle is symmetric about its main diagonal, and (iv) we indicated how to prove that the $n$-th row is a unimodal sequence. Apr 15 Read Section 6.5. We gave two proofs of the binomial theorem. Then we defined trinomial coefficients, gave a formula for them, explained the recursion for them (Pascal's Pyramid), and stated the trinomial theorem. We indicated how to generalize to multinomial coefficients. Quiz 9. Apr 17 Read Section 6.6. Selections with repetitions. We introduced multisets and multichoose coefficients: $MC(n,k) = \left({n \choose k}\right)={{n+k-1}\choose k}$. Apr 19 Read Section 6.3. Inclusion-exclusion. Counting surjective functions. Apr 22 Stirling numbers of the second kind. $S(n,k)$ counts the number of partitions of an $n$-element set into $k$-cells. Quiz 10. Apr 24 More on Stirling numbers of the second kind! We compared results on $C(n,k)$ with the corresponding results on $S(n,k)$. Apr 26 Review sheet for the final! Practice problems! (Hints!) Apr 29 Practice problems! (Some hints!) Quiz 11. May 1 Review for the final!